Fix an imaginary quadratic field $K$, and let $\mathcal{O}_K$ be its ring of integers. A Hermitian modular form of genus 1 (i.e., an automorphic form on $GU(1,1)$) of weight $(k_1,k_2)$ on a congruence subgroup $\Gamma \subset U(1,1)(\mathbb{Z})$ is a function $f \colon \mathfrak{h} \to \mathbb{C}$ satisfying an analytic condition and a transformation law:

$$ f\left(\frac{az+b}{cz+d}\right) = (cz+d)^{k_1}(\overline{c}z+\overline{d})^{k_2} f(z), \qquad \forall \begin{pmatrix} a & b \\ c & d\end{pmatrix} \in \Gamma. $$

Here $\Gamma \subset GU(1,1)(\mathbb{Z})$ is a group of matrices with entries in $\mathcal{O}_K$, so conjugating the coefficients makes sense. At this stage these are very similar to classical modular forms, and in fact they have the same shape $q$-expansions and simply restricting the matrices that are allowed to act on them allows them to be viewed as classical modular forms of weight $k_1+k_2$ on the congruence subgroup $\Gamma \cap SL_2(\mathbb{Z})$.

I have not found anywhere what the (complex) $L$-functions for these things are. For a classical modular cusp form, we have $$ f(z) = \sum_{n=1}^\infty a_n q^n, \qquad \Rightarrow \qquad L(f,s) = \sum_{n=1}^\infty \frac{a_n}{n^s}. $$

Is there a similar formula for what the $L$-function of a hermitian cusp form of genus 1 should be? Is it the same, maybe modulo the Euler factors at infinity? How are the $L$-functions of the hermitian modular form of weight $(k_1,k_2)$ and the classical modular form of weight $k_1+k_2$ related?

Is there a similar relationship between the hermitian modular forms over a CM field $F$ and the Hilbert modular forms over its totally real subfield $F^+$?

Follow up: how should I view the fact that every Hermitian modular form yields a classical modular form, while an elliptic curve over $\mathbb{Q}$ yields an elliptic curve over $K$? If I equate "elliptic curve over $\mathbb{Q}$" to "classical modular form on $\Gamma_0(N)$," and "elliptic curve over $K$" to "hermitian modular form on a similar $\Gamma$," these feel like they go in opposite directions from each other.